分布式波束成形：单天线多中继网络优化策略

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“单天线多中继波束成形的研究着重于优化放大转发中继网络的性能。通过将分布式波束形成问题转化为广义特征向量问题（GEP），可以求得输出SNR的最大值以及对应的特征向量和特征值。文章提出了四类不同的波束形成算法，以适应不同的系统约束和设计目标。” 在无线通信领域，中继网络被广泛用于改善信号传输的质量和覆盖范围。在这种网络中，信号经过中继节点的转发，而不是直接从源节点传送到目的地。单天线多中继波束成形是一种技术，它通过协调多个中继节点的信号传输，以增强信号强度并降低噪声影响，从而提高整个系统的信噪比（SNR）。文章《低复杂度分布式波束形成：实值实现的中继网络》深入探讨了这一主题。首先，作者指出最大化输出SNR可以视为一个GEP，这是一种线性代数问题，其解决方案能够给出最优的波束形成权重。这些权重可以调整，以确保在满足总发射信号功率约束的情况下，获得最高的输出SNR。接着，文章提出了四种不同类型的波束形成算法： 1. **最大输出SNR算法**：该算法在总发射信号功率受限的条件下，寻求最大化输出SNR。这有助于在有限的功率预算下优化通信质量。 2. **最小化总发射信号功率算法**：在保持一定输出SNR水平的前提下，该算法的目标是减少总的信号发射功率，从而节省能源并减少对其他通信频道的干扰。 3. **最小中继节点数算法**：在总信号功率和输出SNR的约束下，该算法旨在找到最少数量的中继节点，以维持所需的通信性能。这有助于降低网络复杂性和成本。 4. **鲁棒算法**：考虑到实际环境中可能存在信道估计误差，此算法设计用于应对这种不确定性，确保在通道估计不准确的情况下仍能保持系统的稳健性能。所有这些算法都采用了实值实现，这意味着它们可以在处理复杂计算时保持较低的计算复杂度，这对于实时通信系统来说是非常重要的。通过这样的优化，中继网络能够在有限的硬件资源和计算能力下实现高效的波束形成策略，提升整体通信效率和可靠性。

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 20, OCTOBER 15, 2013 5039

Low-Complexity Distributed Beamforming for Relay

Networks With Real-Valued Implementation

Lei Zhang, Wei Liu, Senior Member, IEEE,and JianLi, Fellow, IEEE

Abstract—The distributed beamforming problem for amplify-

and-forward relay networks is studied. Maximizing output SNR

(signal-to-noise ratio) for d istributed beamforming can be consid-

ered as a generalized eigenvector problem (GEP) and the prin-

cipal eigenvector and its eigenvalue can be derived with a standard

closed-form solution. In this paper, four classes of beamforming

algorithms are derived based on different design criteria and con-

straints, including maximizing output SNR subject to a constraint

on the total transmitted signal power, minimizing the total trans-

mitted signal power subject to certain level of output S NR, mini-

mizing the relay node number subject to constraints on the total

signal pow er and output SNR, and a robust algorithm to deal with

channel estimation errors. All of the algorithms have a low

com-

putational complexity due to the proposed real-valued implemen-

tation.

Index Terms—Distributed beamforming, relay networks, gener-

alized eigenvector problem, robust algorithm.

I. INTRODUCTION

ISTRIBUTED beamforming (also called collabo rati ve

beamforming ) is a form of cooperative communication

using a relay network consisting of two or more nodes for-

warding the message from a transmitter to an intended receiver

when there is no direct link between them or the link is so

weak that it cannot support the minimum required quality of

service (QoS) [1] –[5]. It can improve th e Q oS when channel

conditions are poor, and the resultant coo per ation diversity

can also provide beneﬁts of increased range and data rate or

improved energy efﬁciency [1], [6], [7]. In general, distributed

beamforming algorithms are divided into three categories:

amplify-and-forward (AF) [6]–[9], decode-and-forward [10],

[9], and compress-and-forwar d [11], [12]. The AF scheme is

of particular interest and h as been studied extensively due to

its sim plicity in both algorithm and implem ent a tion aspects. I n

this paper we will focus on the AF-based one.

Depending on d ifferent design objectives, constraints and

assumptions, various methods have b een proposed based on

the knowledg e of channel state inform ation (CSI). With the

assumption that the source kn ows the direct link (the channel

between itself and the destination), each relay knows its own

Manuscript received October 24, 2012; revised April 14, 2013 and July 10,

2013; accepted July 15, 2013. Date of publication July 26, 2013; date of current

version September 09, 2013. The associate editor coordinating the review of

this manuscript and approving it for publication was Prof. Pascal Larzabal.

L. Zhang is with the Communication Techn ologies Lab, Huawei Technolo-

gies Co., Ltd., China (e-mail: zhanglei211@huawei.com).

W. Liu is with the Communications Research Group, Department of

Electronic and Electrical Engineering, University of Shefﬁeld, U.K. (e-mail:

w.liu@shefﬁeld.ac.uk).

J. Li is with the Dep artment of Electrical and Computer Engineering, Uni-

versity of Florida, Gainesville, FL 32601 USA (e-mail: li@dsp.uﬂ.edu).

gital Object Identiﬁer 10.1109/TSP.2013.2274957

channels and the destinatio n knows all o f the ch annels, the

problem of max imizing output SNR subject to individual power

transmission constraint for both with and witho ut direct link

scenarios was solved in [13]. The distributed beamforming

problem of maximizing output SNR with total and individua l

power constraint s for a three-hop r e lay system was inv est igat ed

in [14]. Instead of maxim izing the ou tpu t S NR, beam forming

schemes based on the minimum mean square error (MMSE)

criterion were studied in [15]–[17]. Based on the second-order

statistics of CSI, two beamforming algorithms, one for m in -

imizing the total transmit power subject to the receiv

er QoS

constraint and the other one for maximizing the recei

ver SNR

subject to two types of power constraints, were prop

osed in

[18]. This work was then extended to two-way commu

nication

systems in [19], and m ultiple peer-to-peer comm

unications

based on a common relay network in [20 ]. Given er

rors in

the dow nlink (from relay nodes to destination

)CSI,arobust

optimization algorithm was deri ved wit

h the consideration that

the uplink (from sou rce to relay nodes)

coefﬁcient could be es-

timated more accurately. Another rob

ust scheme was proposed

in [21] for uplink tran s mission with

each node equipped with

an antenna array.

In this paper, with the knowledge of

instantaneous CSI, we

propose a low-complexity real-va

lued implementation of the

system by introducing a prepro

cessing stage with a set of offset

phase shifts i nto the relay ne

twork. Four problems are then

studied based on this implem

entation. The ﬁrst one is maxi-

mizing output SNR subject t

o a limit on the to tal transmitted

signal power, which can be

considered as a real-valued general-

ized eigenvector proble

m (GEP) w ith its matrix pair consisting

of a special rank-one

signal correlation matrix and a diagonal

noise correlation m

atrix. The GEP is then transformed into an

eigenvector probl

em (EP), where the principal eigenvector and

its eigenvalue can

be derived with a closed-form solution. The

derivation does n

ot involve any eigen-decomposition operation

and avoids the mo

st time consuming part of the algorithm.

The second prob

lem studied is minimizing power consumption

with a constr

aint on a certain lev e l of output SNR. Its optimum

solution ha

s to satisfy the standard structure of the princip a l

eigenvect

or, with a single unknown parameter solv ed by an

iterativ

e method. The third problem is based on a new consid-

eration t

hat the free nodes in a network are limited resources,

and some

of them could enter or exit due to their operation

state a

nd their own consideration such as battery status, and

anewa

lgorithm for minimizing the relay number subject to

apow

er consumption limit and a certain level of output SNR

is p

roposed. Two solutions are derived by setting different

iorities for t he two constraints. The above three problems

e based on perfect instantaneous CSI for both up and dow n

inks. How ever, there may exist estimation errors for both

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